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Learn a new method for measuring asteroid distances using diurnal parallax from a single site. This self-contained technique provides accurate measurements without the need for external data. Discover mathematical insights and practical examples to replicate observations and calculations. Explore the geometry model and the process of estimating geocentric RA rates. See how compensating errors can lead to precise distance estimates from two-night observations.
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Diurnal Parallax Determinations of Asteroid DistancesUsing Only Backyard Observations from a Single Station Eduardo Manuel Alvarez Observatorio los Algarrobos, Salto, Uruguay Robert Buchheim Altimira Observatory, Coto de Caza, CA USA SAS XXXI Symposium – Big Bear Lake, CA, USA – May 23-24, 2012
Subject • New method for measuring asteroid parallax from a single site • simple, accurate, self-contained • three mathematical insights • Technical base supporting the new method • Some practical examples
Single-site “Diurnal Parallax” Earth’s rotation provides a “baseline” so that a single observer can measure the parallax. Parallax angle φ(t) = [ RAtopocentric – RAgeocentric ] cosδ
Objectives: Demonstrate measurement of distance to asteroid single-site diurnal parallax “Self-Contained” method All necessary data observable from single site no need for ephemeris or external data Challenges: Infer “geocentric” position from “topocentric” observations. Asteroids move … rapidly! “back-out” secular motion, leaving only parallax motion Accuracy vs. Simplification: measurements & calculations Objectives & Challenges
μ ν Z= R∙sinδ D R observer δ z R∙cosδ y α x Geometry of Model • Geocentric RA = Topocentric RA when target is at transit • use sequential transits to measure geocentric RA rate (ν) • + Assume: • Geocentric distance ≈ constant • geocentric Dec rate ≈ topocentric Dec rate • Secular Geocentric rates are (approx.) constant
RAgeo rate estimated by RA(t2) – RA(t1) ν≈ ------------------- t2 – t1 Estimate Geocentric RA ratefrom Consecutive Transit Observations night #2 “RA rate ν” (measured) night #1 RA position time
Found parallax = 7.52 arc-sec Found distance = 0.997 AU True distance = 0.973 AU error = 2.5% Pretty good! Example: 8106 Carpino parallax, arc-sec elapsed time, hr
Estimating Geocentric RA rateUsing Consecutive Transit Observations… but not at opposition night #2 night #1 RA position “true” geocentric RA motion (not measurable) time
Constant-RA-rate linear approximation of “true” RA(t) curve night #2 culmination “RA rate ν” (measured) night #1 culmination RA position “true” geocentric RA motion (not measurable) time
On night #1, νover-estimates the RA rate … Compensating errors night #2 culmination night #1 culmination RA position time
On night #1, ν over-estimates the RA rate … On night #2, νunder-estimates the RA rate Compensating errors night #2 culmination night #1 culmination RA position time
Wonderfully, Using both nights, the errors in calculated parallax / distance tend to cancel out, resulting in a very accurate (average) distance estimate. Compensating errors night #2 culmination night #1 culmination RA position time
Found parallax = 2.8 arc-sec Found distance = 2.67 AU True distance = 2.545 AU Error = 5.1% Distant target: 414 Liriope parallax, arc-sec elapsed time, hr
Found parallax = 145 arc-sec Found distance = 0.052 AU True distance = 0.049 AU Error = 6.1% Fast-mover, Near-Earth Asteroid (162421) 2000 ET70 parallax, arc-sec elapsed time, hr
Each night, many data points to fit sine-curve Two nights, two short sets of observations each night: early in night at transit How many data points do you really need? “Four-Point Shortcut”
Each night, many data points to fit sine-curve Two nights, two short sets of observations each night: early in night at transit How many data points do you really need? With good astrometric images, “four-point shortcut” gives excellent distance estimates “Four-Point Shortcut”
Conclusions:Measurement of Asteroid Parallax • A feasible “backyard” project • Modest equipment, standard software • Replicate historically-important observations and calculations: • observe diurnal parallax effect • measure distance to solar-system object • determine scale of Solar System (“Astronomical Unit”) • Mathematical insights simplify: • Observations early in evening, and at transit • then go to bed … • Two consecutive nights provide sufficient data
